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OCCAMS: A Multi-Task Learning-Based Investigation of the CMB Power Spectra

Ankit Mandal, Rahul Shah, Md. Wali Hossain, Somasri Sen, Anjan A Sen, Supratik Pal

arXiv:2610.40415v1astro-ph.COastro-ph.IMcs.LG

TL;DR

How many latent variables capture the full CMB temperature and polarisation spectra, and how do they relate to standard cosmological parameters? The paper trains VAEs on simulated spectra without building a cosmological model into the network, finding that six latent variables are best or close to best in most studied cases and that τ and A_s remain nearly independent in the latent representation.

  • Problem

    The number of independent degrees of freedom needed to describe full CMB spectra without a predefined physical parametrisation, and their relation to ΛCDM parameters, remain open questions.

    Source: §1 · p. 1, §5 · p. 10

  • Method

    The study trains VAEs on CLASS-generated spectra and uses a soft disentanglement penalty to compare free latent variables with ΛCDM parameters after training.

    Source: §1 · p. 2, §3.2 · p. 4, §3.3.3 · p. 6

  • Results

    L=6 is best or within ln B≃1 of the best in five of six cases, while τ and A_s latent variables share normalised MI of at most 0.03 across all 18 models.

    Source: §6 · p. 13

  • Takeaways & Limitations

    The results support a compact, data-driven parametrisation of CMB temperature and polarisation spectra, with latent variables mapping more clearly to ω_b, τ, ω_cdm and h than to A_s and n_s.

    Source: §6 · p. 13

    Although the network has no cosmological model built in, its training spectra are generated within ΛCDM.

    Source: §2 · p. 3

Abstract

from arXiv · show

How many latent variables are needed to fit the cosmic microwave background (CMB) temperature and polarisation power spectra when a neural network is given no cosmological model, and to what extent do they map onto the standard cosmological parameters? We use multi-task learning with variational autoencoders trained on CLASS-generated spectra (TT, TE, EE) over $2\le\ell\le2500$. No cosmological model is built into the network, and the number of latent variables, $L$, is varied over $L=5$, $6$ and $7$. Their fits are compared to the Planck 2018 data using $\hat{\chi}^2$, AIC, BIC and Bayesian evidence. Their correspondence with cosmological parameters is measured with normalised mutual information post-training. The loss weights are set within a single training run by uncertainty weighting, with the weight of the disentanglement term either fixed (semi-automated, SA) or learned (completely automated, CA). The reconstruction loss is split over multipole ranges in three configurations that separate the roles of temperature and polarisation and of the low- and high-$\ell$ ranges. In four of the six combinations of configuration and strategy, the fit improves from $L=5$ to $L=6$ and not beyond, and $L=6$ is the best, or close to the best, in five of the six cases. So, six latent variables are needed, as many as the free parameters of the standard, model-dependent approach. SA gives clearer disentanglement, while CA needs no a priori choice of the disentanglement weight and has the higher evidence at $L=5$ and $6$: the two strategies are points on a trade-off between interpretability and fit to the data. Notably, the latent variables for $\tau$ and $A_s$ share almost no information in any model, consistent with large-scale polarisation separating the two, as in standard analyses. A deep-learning approach can thus give a compact, data-driven parametrisation of the CMB spectra.

1 INTRODUCTION

CMB analyses commonly use a fixed ΛCDM parameterisation, although the data constrain parameter combinations more strongly than individual parameters. This work tests whether an unconstrained latent representation can capture the full temperature and polarisation spectra and how its latent variables relate to ΛCDM parameters.

1 INTRODUCTION

  • CMB data constrain combinations of ΛCDM parameters more strongly than individual parameters, while rigid parameterisations can yield complex degenerate posteriors and prior-volume effects.The six ΛCDM parameters are defined theoretically and for physical interpretability.Source: §1 · pp. 1–2
  • Earlier VAE studies found five latent variables sufficient for spectra at ℓ≥30, but did not establish a clean one-to-one mapping to ΛCDM parameters.The prior studies used TT alone or parameter-conditioned TT, TE and EE spectra; the full low-ℓ range remained untested.Source: §1 · p. 2
  • The study asks how many latent variables fit full TT, TE and EE spectra over 2≤ℓ≤2500 and how they map to the six ΛCDM parameters, without embedding a cosmological model in the network.The reconstruction loss is split into three configurations to assess temperature, polarisation, and low- versus high-ℓ contributions.Source: §1 · pp. 2–3
  • The latent variables are not assigned parameters in advance; a soft disentanglement penalty encourages distinct physical directions, and the mapping is inferred after training.The penalty permits a latent to depend on up to two parameters rather than requiring a strict one-to-one mapping.Source: §1 · p. 3
  • Uncertainty weighting learns loss-term weights in one run; SA fixes the disentanglement weight at 5, whereas CA learns it from the data.The strategies compare stronger manually set disentanglement with fully automated weighting.Source: §1 · p. 3

2 DEEP GENERATIVE MODELLING AND VARIATIONAL AUTOENCODERS

A VAE compresses each input spectrum into a probabilistic latent representation and reconstructs it, balancing reconstruction against regularisation. The training spectra are generated with ΛCDM as a simulator, not as information built into the network.

2 DEEP GENERATIVE MODELLING AND VARIATIONAL AUTOENCODERS

  • A VAE encodes an input spectrum as a Gaussian latent distribution, samples a latent vector, and decodes it to reconstruct the spectrum.The encoder outputs means and standard deviations; the decoder reconstructs from the stochastic sample.Source: §2 · p. 3, Figure 1 · p. 4
  • The reconstruction term measures mean squared error between rescaled predicted and training spectra, while the KL term encourages the learned latent distribution to resemble a predefined prior.The training spectra are the encoder inputs; ΛCDM supplies realistic simulated spectra, but the network sees only those spectra and is not told their generating parameters.Source: §2 · p. 3
  • Choosing the KL weight β balances reconstruction accuracy and latent disentanglement, but the optimal value is unknown and manual scans over the loss weights can be computationally prohibitive.The modified objective introduces further hyperparameters beyond β.Source: §2 · p. 3

3 METHODOLOGY

The spectra are compressed by a one-dimensional convolutional VAE into a fixed-size probabilistic latent space, then reconstructed through transposed convolutions. Separate models are trained with five, six, or seven latent dimensions.

3 METHODOLOGY

  • The 1D convolutional VAE consists of an encoder, stochastic latent bottleneck, and decoder that maps high-dimensional spectra into a continuous low-dimensional probabilistic space.The pipeline is implemented in OCCAMS.Source: §3.1 · p. 4
  • Three convolutional blocks encode the three input spectra channels and project extracted features to a 2L vector for the latent means and log-variances.The log-variance is clamped between −10 and 10 for numerical stability.Source: §3.1 · p. 4
  • The latent representation is an L-dimensional diagonal-covariance Gaussian, with separate models at L=5, 6, and 7; the decoder upsamples sampled latents through transposed convolutions to three output spectra.The reparameterisation trick allows gradients to pass through stochastic sampling.Source: §3.1 · p. 4

3.2 Training Data

CLASS generates the simulated TT, TE and EE training spectra across 2≤ℓ≤2500, with parameter combinations selected by Latin hypercube sampling. The spectra are transformed and divided into training, validation, and test sets.

3.2 Training Data

  • CLASS maps sampled cosmological parameters to temperature and polarisation spectra, used only as training examples; the ranges are covered evenly without imposed parameter correlations.The spectra span 2≤ℓ≤2500 and are split into 80% training, 10% validation, and 10% test data.Source: §3.2 · p. 4
  • Each spectrum is converted to Dℓ, divided by a reference spectrum, log-transformed, and normalised; a vertical offset makes sign-changing TE spectra compatible with the logarithm.The VAE output undergoes the same steps in reverse to recover predicted Dℓ spectra.Source: §3.2 · pp. 4–5
  • The parameter ranges span approximately 10σ around Planck 2018 baseline best-fit values, with τ restricted to a lower bound of 0.01 and the upper h bound broadened to include the SH0ES measurement.These are the stated exceptions to the general range choice.Source: Table 1 · p. 5

3.3 Loss Function

The loss design addresses the temperature-only A_s–τ degeneracy by incorporating large-scale polarisation and separating reconstruction errors across spectra and multipole ranges. A soft disentanglement penalty encourages distinct physical directions in the latent space without fixing its size or assigning variables to parameters.

3.3.1 Temperature Anisotropies (TT)

  • Piras et al. (2025) found five latent variables sufficient for TT at ℓ≥30, with the missing sixth degree of freedom linked to the A_s–τ degeneracy.In this work, reconstruction and KL weights, and in CA also the disentanglement weight, are learned automatically in one training run rather than tuned by hand.Source: §3.3.1 · p. 5

3.3.2 Including Polarisation (TE, EE)

  • At ℓ≲15, the EE reionisation bump scales approximately as C_ℓ^EE ∝ A_sτ², providing separate information on τ from the high-ℓ A_se^-2τ combination.The reconstruction loss is split into low-ℓ (ℓ≤14), high-ℓ (ℓ≥15), and full-range MSEs so the low-ℓ polarisation signal is not outweighed by the many high-ℓ multipoles.Source: §3.3.2 · p. 5
  • Three reconstruction-loss configurations group temperature and polarisation errors and low- and high-ℓ ranges differently; one gives low-ℓ temperature and polarisation separate weights.In lowTTTEEE+full, the low-ℓ range appears in both the low-ℓ and full-range terms.Source: §3.3.2 · p. 5

3.3.3 Differentiable Disentanglement Loss Formulation

  • Reconstruction and KL losses alone do not tie latent coordinates to physical quantities, so the loss adds a penalty to support interpretation and direct reading of latent posteriors.The anchor describes the motivation for the penalty; its parameter dependence and soft constraints are specified separately.Source: §3.3.3 · p. 6
  • The differentiable disentanglement penalty encourages each latent to depend on as few cosmological parameters as possible and different latents on different parameters, while allowing up to two parameters per latent.It is a soft penalty computed from training-sample latents and parameters; it does not set L or assign a parameter to a latent, and uses squared Pearson correlation rather than the later normalised mutual information.Source: §3.3.3 · p. 6

3.4 Algorithm: Automated Hyperparameter Tuning

The method learns relative loss weights during one training run by treating each loss term as a task with a trainable uncertainty parameter. The two strategies differ in whether the disentanglement weight is fixed or learned.

3.4 Algorithm: Automated Hyperparameter Tuning

  • Uncertainty weighting adapts loss weights during one training run, avoiding repeated trial-and-error runs to choose the many weights.The loss terms are treated as separate tasks, with trainable parameters optimized by gradient descent alongside network parameters.Source: §3.4 · p. 6
  • Each loss term receives a trainable σ_i, with weight w_i = 1/(2σ_i²); the log σ_i term prevents all weights from collapsing to zero.For fixed network parameters, the loss is minimized at σ_i² = L_i, so terms with larger persistent losses receive smaller weights; extending the Gaussian-likelihood weighting to KL and disentanglement is heuristic.Source: §3.4 · p. 6
  • The figure shows Planck 2018 best-fit ΛCDM TT, TE, and EE spectra, with each configuration in paired SA and CA columns and latent dimensions L=5 (blue), L=6 (orange), and L=7 (green).Source: §3.4 · p. 7

3.4.1 Treatment of the Disentanglement Loss Weight

  • SA fixes w_d at 5.0 and excludes disentanglement from uncertainty weighting, whereas CA learns w_d alongside the other loss weights.The disentanglement term is the only loss term that uses cosmological parameter information, and acts as a soft penalty encouraging separate latent features.Source: §3.4.1 · p. 7

4 RESULTS

Across the six configuration–strategy cases, L=6 is the best or close to the best fit in five, while adding a seventh latent variable brings no significant gain. Normalised mutual information shows that latent-variable interpretability varies by strategy, although the variables matched to τ and A_s share almost no information in every model.

4.1 Reconstruction

  • L=5 reconstructions are generally less accurate than L=6 and L=7, whose residuals are similar.Residuals are largest at spectrum peaks and troughs.Source: §4.1 · p. 8

4.2 Interpreting the Latent Space

  • Normalised MI measures information shared by latent variables and ΛCDM parameters, while the disentanglement score also penalises mixed or spread-out associations and unmatched parameters.MI is estimated from Gaussian-mixture fits to each variable pair; the score matches pairs with the Hungarian algorithm and retains pairs at or above 0.10.Source: §4.2 · p. 8
  • Only three models cleanly assign all six parameters, while matched variables for τ and A_s share at most 0.03 normalised MI in every model.The clean assignments occur in SA lowTTTEEE+full at L=6 and 7, and SA lowTTTEEE+high at L=7.Source: §4.2 · p. 8
  • SA has a higher mean disentanglement score than CA, 0.58 versus 0.40; the strategies differ only in whether the disentanglement weight is fixed or learned.The score gap varies by configuration and latent dimension.Source: §4.2 · p. 8

4.3 Latent Space Posteriors

  • The posterior fit compares decoder spectra with Planck 2018 using Plik_lite, then evaluates reduced χ², AIC, BIC and Bayesian evidence.The posterior is sampled with emcee using 64 walkers; the comparison uses TT over 2≤ℓ≤2500 and TE/EE over 30≤ℓ≤1996.Source: §4.3 · pp. 8, 10
  • L=6 is best or within ln B≈1 of the best in five of six cases; L=7 gives no significant gain over L=6.The evidence strongly disfavors L=5 in four cases; the comparison between L=5 and L=6 is more relevant than comparisons involving L=7.Source: Table 3 · p. 10, §4.3 · p. 10

5 DISCUSSION

The discussion finds that six latent variables are most consistently supported for the full CMB spectra, with large-scale polarisation separating the latent representations of τ and A_s. The mappings and the fit–interpretability balance vary across models and loss strategies.

5.1 The Intrinsic Dimensionality of the CMB

  • L=6 is best or within ln B≃1 of the best in five of six cases; L=7 gives no significant gain, while two lowTTTEEE+full cases favour or tie L=5.The four comparison measures favour L=6 over L=5 in four cases; for semi-automated models, the preferred dimension can depend on the measure.Source: §5.1 · p. 11

5.2 Breaking of the 𝑨s–𝝉Degeneracy

  • The τ and A_s latent variables share at most 0.03 normalised MI across all 18 models, consistent with large-scale EE polarisation constraining τ separately.The separation appears without assigning latent variables to parameters in advance and agrees with the role of the reionisation bump in standard analyses.Source: §5.2 · p. 11

5.3 What the Latent Variables Capture

  • The clearest parameter mappings are usually ω_b, τ, ω_cdm and h; n_s and A_s are most often at or below the 0.10 disentanglement threshold.The discussion relates stronger mappings to distinctive spectral features and notes overlapping effects for A_s with n_s and, more weakly, ω_cdm with h.Source: §5.3 · pp. 11–12
  • The extra degree of freedom relative to ℓ≥30 analyses is associated with large-scale polarisation, which constrains τ separately from A_s.At ℓ≥30, earlier analyses found five latent variables, where A_s and τ enter through A_s e^-2τ.Source: §5.3 · p. 12

5.4 Semi-Automated vs Completely Automated Disentanglement

  • In SA, disentanglement scores rise with L, but the best fit need not coincide with the highest score, and choosing L by fit alone need not yield the most interpretable space.Three SA models with L≥6 achieve a clean one-to-one assignment of all six parameters.Source: §5.4 · p. 12
  • SA yields clearer disentanglement, while CA has higher evidence at L=5 and 6 and a more consistent preference for L; together they trade interpretability against fit.The learned CA disentanglement weight is 0.45–0.81, below SA’s fixed 5.0; the comparison suggests weaker alignment pressure can accompany better Planck fit.Source: §5.4 · p. 12

6 CONCLUSIONS AND OUTLOOK

The paper tests VAEs on CLASS-generated TT, TE and EE spectra against Planck data, varying latent dimension and loss configuration without embedding a cosmological model in the network. It finds that six latent variables are most consistently supported, while the strategies differ in disentanglement and fit.

6 CONCLUSIONS AND OUTLOOK

  • The study trains 18 VAEs on CLASS TT, TE and EE spectra over 2≤ℓ≤2500, compares fits with Planck data, and assesses mappings using normalised MI.Uncertainty weighting sets reconstruction and KL weights, and also the disentanglement weight for CA, within a single training run.Source: §6 · p. 12
  • Across 18 models, the τ and A_s latent variables share at most 0.03 normalised MI, consistent with large-scale polarisation separating the parameters.The latent variables are not assigned to parameters in advance.Source: §6 · p. 13
  • Six latent variables are best or within ln B≃1 of the best in five of six cases; four cases improve from L=5 to L=6, and L=7 adds no significant gain.The two lowTTTEEE+full cases differ: SA ties L=5 and L=6, while CA fits best at L=5.Source: §6 · p. 13
  • SA gives clearer disentanglement, while CA avoids fixing the disentanglement weight in advance and has higher evidence at L=5 and 6.The conclusion frames the strategies as a trade-off between interpretability and fit; CA remains preferred when each strategy is compared at its best L.Source: §6 · p. 13
  • The latent variables map more clearly to ω_b, τ, ω_cdm and h than to A_s and n_s, yielding a compact parametrisation of the spectra.The network and latent space contain no built-in cosmological model, although the training spectra are generated within ΛCDM.Source: §6 · p. 13
  • The authors identify higher precision and wider multipole coverage as future tests, including whether improved large-scale polarisation changes τ–A_s separation.They also state that the VAE architecture and training strategy are not final and that statistical robustness can be improved.Source: §6 · p. 13

APPENDIX A: CONSTRUCTION OF Ldis

Appendix A constructs a parameter-to-latent information matrix from standardised batches and uses it to penalise diffuse or overlapping latent representations. The three loss components encourage concentrated, distinct mappings while allowing two-parameter degeneracies when needed.

APPENDIX A: CONSTRUCTION OF Ldis

  • Standardised latent representations Z and parameter labels Θ are used to compute squared Pearson correlations between every latent dimension and physical parameter.The batch contains N samples; L denotes latent dimension and P the number of parameters.Source: Appendix A · p. 14
  • Column-wise normalisation turns the squared-correlation matrix into M, whose columns sum to one and represent each latent variable’s fractional information capacity across parameters.The stabilising constant is ε=10^-8.Source: Appendix A · p. 14
  • The disentanglement objective combines three terms, with hyperparameters controlling the trade-off between orthogonality and strictness.A small β′ acts as a soft regulariser against unnecessary use of the two-parameter allowance.Source: Appendix A · pp. 14–15
  • The top-2 purity loss penalises a latent variable that spreads information across more than two parameters, while strict purity discourages using the two-parameter allowance unnecessarily.Strict purity is intended to favour one-to-one mappings, retaining the allowance when physically necessary to lower reconstruction error.Source: Appendix A · pp. 14–15
  • The collision loss penalises off-diagonal overlap in O=M^T M so separate latent dimensions do not track the same parameter.Figure A1 depicts the calculation from standardised parameter and latent batches through the purity, strict and collision terms.Source: Appendix A · p. 15, Figure A1 · p. 15

APPENDIX B: LOSS FUNCTION WEIGHTS

The appendix reports the loss weights after training across all 18 models, noting that low-ℓ reconstruction terms receive substantially larger weights than high-ℓ terms.

APPENDIX B: LOSS FUNCTION WEIGHTS

  • Low-ℓ loss weights are roughly two orders of magnitude larger than high-ℓ weights, while β ranges from 0.15 to 0.26 across all models.Table B1 defines the low- and high-ℓ reconstruction weights, the KL weight β and the disentanglement weight w_d.Source: Appendix B · p. 15

APPENDIX C: DISENTANGLEMENT SCORE DEPENDENCE ON MI THRESHOLD

The appendix tests how the disentanglement score changes with the MI threshold, finding that top-model rankings are fairly robust while lower-model rankings can shift.

APPENDIX C: DISENTANGLEMENT SCORE DEPENDENCE ON MI THRESHOLD

  • The three highest-scoring models at the nominal threshold remain the top three up to an MI threshold of 0.17.Above 0.17, no model exceeds a score of 0.58.Source: Appendix C · p. 15
  • CA scores lose more between thresholds 0.05 and 0.10 than SA scores, while lower-scoring models change order as the threshold varies.The passage cautions against over-interpreting differences among lower-scoring models.Source: Appendix C · p. 15
  • Below about 0.04, all model scores remain at their plateau values.Source: Appendix C · p. 15

APPENDIX D: MCMC CONTOURS

The appendix presents latent-variable posterior contours for the lowTTTEEE+high models and cautions that latent dimensions are not individually comparable across panels.

APPENDIX D: MCMC CONTOURS

  • For L=6 and L=7, some latent-variable pairs are almost perfectly correlated or anti-correlated, so the data constrain their combination rather than each variable separately.Source: Appendix D · p. 15
  • The latent-variable numbering, sign, and scale are arbitrary, so panels cannot be compared latent by latent.Source: Appendix D · p. 15
  • The D1, D2, and D3 panels show posterior distributions for L=5, L=6, and L=7, respectively, with 68% and 95% contours.Each figure compares SA in the top panel with CA in the bottom panel.Source: Figure D1 · p. 16, Figure D2 · p. 17, Figure D3 · p. 17
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